[9] PML blueprints can be derived systematically for general geometries through a two step process. In the first step, a complexification of the metric of space is employed to map ordinary solutions of Maxwell's equations continuously into non-Maxwellian fields which exhibit a modified behavior (e.g., exponential decay) inside the PML [*Teixeira and Chew*, 1999a]. This complexification can be carried out by an analytic continuation of the spatial coordinates [*Teixeira and Chew*, 1998]. In the second step, another field mapping is employed. This second mapping transforms the non-Maxwellian (analytic continued) fields into a third set of fields, which are Maxwellian [*Teixeira and Chew*, 1999a]. This latter map explores the metric invariance of Maxwell's equations (in the sense of *Deschamps* [1981], *Teixeira and Chew* [1999b], and *Bossavit* [2001]), to recast the complexification of the metric as a change on the constitutive parameters. The end result is a set of Maxwell's equations in anisotropic media, where both the permeability and permittivity tensors are frequency dispersive and depend on the local metric coefficients of the PML surface (or, equivalently, the local radii of curvature of the PML).

[10] To describe these tensors, we attach a local orthogonal curvilinear coordinate system (ξ_{1}, ξ_{2}, ξ_{3}) to a point *P* on the PML surface, where ξ_{1}, ξ_{2} are tangent coordinates to the surface and ξ_{3} is the normal coordinate, such that ξ_{3} = 0 represents the PML interface itself and ξ_{3} > 0 represents points *inside* the PML. Using the convention *e*^{−iωt}, the PML blueprints for a doubly curved surface are written as

with the PML tensor [*Teixeira and Chew*, 1998]

In the above, *s* is the so-called complex stretching variable [*Chew and Weedon*, 1994], which has a frequency dependence on the form *s*(ω, ξ_{3}) = *a*(ξ_{3}) + *i* σ(ξ_{3})/ω, with *a*(ξ_{3}) ≥ 1 and σ(ξ_{3}) ≥ 0 in the PML (ξ_{3} > 0) (other functional dependencies of *s* in terms of ω are possible as long as they lead to an absorptive behavior). The factors _{i} and *h*_{i}, *i* = 1, 2 are the stretched and nonstretched local metric coefficients, respectively, given by *h*_{i} = *r*_{i}/*r*_{0i} and = _{i}/*r*_{0i}, *i* = 1, 2, and *h*_{3} = 1, where *r*_{0i}(ξ_{1}, ξ_{2}), *i* = 1, 2, are the local principal radii of curvature at the point (ξ_{1}, ξ_{2}, 0). The radii of curvature are defined from the *outside* of the surface. Therefore, they are positive over concave surfaces and negative over convex surfaces. Moreover, *r*_{i} = *r*_{0i} + ξ_{3} and _{i} = *r*_{0i} + _{3}*i* = 1, 2, with

being the analytic continuation of the coordinate ξ_{3}. The unit vectors _{i}, *i* = 1, 2, are tangential to *S* at *P* along the principal lines of curvature, and = _{1} × _{2} is the unit vector normal to *S* at this point. In terms of the local coordinates ξ_{1}, ξ_{2}, ξ_{3} of a local orthogonal system, we write _{i} = (∂**r**/∂ξ)/∣∂**r**/∂ξ_{i}∣, *i* = 1, 2, where **r** is the position vector, and analogously for . The imaginary part of the complex stretching variable *s* is responsible for the loss mechanism inside the PML region.

[11] The significance attached to equation (1) is that reflectionless absorption of incident waves on an curved surface can be obtained through by a hypothetical medium having properly chosen constitutive tensors. Strictly speaking, however, it is clear that the PML tensors given by equation (1) cannot represent the frequency behavior of real materials for all frequencies and, as a result, no physical significance can be attached to the absorber blueprint in equation (1) other than the (desired) constitutive behavior (objective function) to be *approximated* over some *finite*, preassigned bandwidth of interest. (For instance, in the limit ω ∞ (in practice, this would correspond to the far ultraviolet for light element materials and X-ray frequencies for heavy element materials), a real material exhibits the following limiting approximate behavior for the permittivity (the fine structure is ignored for simplicity.) This has been the approach taken, for instance, in the work of *Ziolkowski* [1997a, 1997b] for physically realizable planar PMLs,

where ω_{p}^{2} = *Ne*^{2}/*m*ϵ_{0}, *N* is the number of atoms per unit volume, *e* and *m* are the electron charge and mass, respectively, and γ is a damping constant related to the collision rate. This approximate dependence is not exhibited by equation (1). In terms of the permeability, slowly decaying μ_{ij} terms which decreases as 1/ω may be present up to optical frequencies for some anisotropic media such as ferromagnets. Also, some polarization modes related to the core electrons in the atoms may give rise to variations on the ϵ(ω) behavior other than above even at X-ray frequencies, but this is not germane to the main discussion here.)